photo paper or other thick, good quality paper. Avoid using regular multi-purpose computer paper, since it's thin and the ink will make it wrinkle. Matte paper is always good, since it's easier for printing your resume on the back of your headshot
yes, my opinion regular computer paper is not good one. Matte paper produces nice quality prints but does not have that vibrant color finish. Matte paper is not vulnerable to being marred by fingerprints and it produces a non-glare photo.
I recently had an opportunity to sit down with a production manager and this question actually came up in the discussion. Headshots need to be taken seriously and printing them on normal paper usually doesn't get the job done for a major production. They managers expect professional quality presentation from professional quality people. Just take a look at some online reviews, people who have the best experiences get gigs at professional companies. Have you guys ever seen those kinds of reviews, the ones on angies list?